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Journal Cover Physica D: Nonlinear Phenomena
  [SJR: 1.049]   [H-I: 102]   [3 followers]  Follow
    
   Hybrid Journal Hybrid journal (It can contain Open Access articles)
   ISSN (Print) 0167-2789
   Published by Elsevier Homepage  [3089 journals]
  • Hyperbolic periodic orbits in nongradient systems and small-noise-induced
           metastable transitions
    • Authors: Molei Tao
      Pages: 1 - 17
      Abstract: Publication date: 15 January 2018
      Source:Physica D: Nonlinear Phenomena, Volume 363
      Author(s): Molei Tao
      Small noise can induce rare transitions between metastable states, which can be characterized by Maximum Likelihood Paths (MLPs). Nongradient systems contrast gradient systems in that MLP does not have to cross the separatrix at a saddle point, but instead possibly at a point on a hyperbolic periodic orbit. A numerical approach for identifying such unstable periodic orbits is proposed based on String method. In a special class of nongradient systems (‘orthogonal-type’), there are provably local MLPs that cross such saddle point or hyperbolic periodic orbit, and the separatrix crossing location determines the associated local maximum of transition rate. In general cases, however, the separatrix crossing may not determine a unique local maximum of the rate, as we numerically observed a counter-example in a sheared 2D-space Allen–Cahn SPDE. It is a reasonable conjecture that there are always local MLPs associated with each attractor on the separatrix, such as saddle point or hyperbolic periodic orbit; our numerical experiments did not disprove so.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.10.001
      Issue No: Vol. 363 (2017)
       
  • A computational exploration of the McCoy–Tracy–Wu solutions of the
           third Painlevé equation
    • Authors: Marco Fasondini; Bengt Fornberg; J.A.C. Weideman
      Pages: 18 - 43
      Abstract: Publication date: 15 January 2018
      Source:Physica D: Nonlinear Phenomena, Volume 363
      Author(s): Marco Fasondini, Bengt Fornberg, J.A.C. Weideman
      The method recently developed by the authors for the computation of the multivalued Painlevé transcendents on their Riemann surfaces (Fasondini et al., 2017) is used to explore families of solutions to the third Painlevé equation that were identified by McCoy et al. (1977) and which contain a pole-free sector. Limiting cases, in which the solutions are singular functions of the parameters, are also investigated and it is shown that a particular set of limiting solutions is expressible in terms of special functions. Solutions that are single-valued, logarithmically (infinitely) branched and algebraically branched, with any number of distinct sheets, are encountered. The algebraically branched solutions have multiple pole-free sectors on their Riemann surfaces that are accounted for by using asymptotic formulae and Bäcklund transformations.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.10.011
      Issue No: Vol. 363 (2017)
       
  • Synchronization of impacting mechanical systems with a single constraint
    • Authors: Michael Baumann; J.J. Benjamin Biemond; Remco I. Leine; Nathan van de Wouw
      Pages: 9 - 23
      Abstract: Publication date: 1 January 2018
      Source:Physica D: Nonlinear Phenomena, Volume 362
      Author(s): Michael Baumann, J.J. Benjamin Biemond, Remco I. Leine, Nathan van de Wouw
      This paper addresses the synchronization problem of mechanical systems subjected to a single geometric unilateral constraint. The impacts of the individual systems, induced by the unilateral constraint, generally do not coincide even if the solutions are arbitrarily ‘close’ to each other. The mismatch in the impact time instants demands a careful choice of the distance function to allow for an intuitively correct comparison of the discontinuous solutions resulting from the impacts. We propose a distance function induced by the quotient metric, which is based on an equivalence relation using the impact map. The distance function obtained in this way is continuous in time when evaluated along jumping solutions. The property of maximal monotonicity, which is fulfilled by most commonly used impact laws, is used to significantly reduce the complexity of the distance function. Based on the simplified distance function, a Lyapunov function is constructed to investigate the synchronization problem for two identical one-dimensional mechanical systems. Sufficient conditions for the uncoupled individual systems are provided under which local synchronization is guaranteed. Furthermore, we present an interaction law which ensures global synchronization, also in the presence of grazing trajectories and accumulation points (Zeno behavior). The results are illustrated using numerical examples of a 1-DOF mechanical impact oscillator which serves as stepping stone in the direction of more general systems.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.10.002
      Issue No: Vol. 362 (2017)
       
  • Monotonicity, oscillations and stability of a solution to a nonlinear
           equation modelling the capillary rise
    • Authors: Mateusz
      Abstract: Publication date: 1 January 2018
      Source:Physica D: Nonlinear Phenomena, Volume 362
      Author(s): Łukasz Płociniczak, Mateusz Świtała
      In this paper we analyse a singular second-order nonlinear ODE which models the capillary rise of a fluid inside a tubular column. We prove global existence, uniqueness and find several approximations along with the asymptotic behaviour of the solution. Moreover, we are able to find a critical value of the nondimensional parameter for which the solution exhibits a transition in its behaviour: from being monotone to oscillatory. This is an analytical rigorous proof of the experimentally and numerically confirmed phenomenon.

      PubDate: 2017-12-12T12:38:45Z
       
  • Synchronisation under shocks: The Lévy Kuramoto model
    • Authors: Dale Roberts; Alexander C. Kalloniatis
      Abstract: Publication date: Available online 11 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Dale Roberts, Alexander C. Kalloniatis
      We study the Kuramoto model of identical oscillators on Erdős-Rényi (ER) and Barabasi–Alberts (BA) scale free networks examining the dynamics when perturbed by a Lévy noise. Lévy noise exhibits heavier tails than Gaussian while allowing for their tempering in a controlled manner. This allows us to understand how ‘shocks’ influence individual oscillator and collective system behaviour of a paradigmatic complex system. Skewed α -stable Lévy noise, equivalent to fractional diffusion perturbations, are considered, but overlaid by exponential tempering of rate λ . In an earlier paper we found that synchrony takes a variety of forms for identical Kuramoto oscillators subject to stable Lévy noise, not seen for the Gaussian case, and changing with α : a noise-induced drift, a smooth α dependence of the point of cross-over of synchronisation point of ER and BA networks, and a severe loss of synchronisation at low values of α . In the presence of tempering we observe both analytically and numerically a dramatic change to the α < 1 behaviour where synchronisation is sustained over a larger range of values of the ‘noise strength’ σ , improved compared to the α > 1 tempered cases. Analytically we study the system close to the phase synchronised fixed point and solve the tempered fractional Fokker–Planck equation. There we observe that densities show stronger support in the basin of attraction at low α for fixed coupling, σ and tempering λ . We then perform numerical simulations for networks of size N = 1000 and average degree d ̄ = 10 . There, we compute the order parameter r as a function of σ for fixed α and λ and observe values of r ≈ 1 over larger ranges of σ for α < 1 and λ ≠ 0 . In addition we observe drift of both positive and negative slopes for different α and λ when native frequencies are equal, and confirm a sustainment of synchronisation down to low values of α . We propose a mechanism for this in terms of the basic shape of the tempered stable Lévy densities for various α and how it feeds into Kuramoto oscillator dynamics and illustrate this with examples of specific paths.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.12.005
       
  • Computing Evans functions numerically via boundary-value problems
    • Authors: Blake Barker; Rose Nguyen; Björn Sandstede; Nathaniel Ventura; Colin Wahl
      Abstract: Publication date: Available online 9 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Blake Barker, Rose Nguyen, Björn Sandstede, Nathaniel Ventura, Colin Wahl
      The Evans function has been used extensively to study spectral stability of travelling-wave solutions in spatially extended partial differential equations. To compute Evans functions numerically, several shooting methods have been developed. In this paper, an alternative scheme for the numerical computation of Evans functions is presented that relies on an appropriate boundary-value problem formulation. Convergence of the algorithm is proved, and several examples, including the computation of eigenvalues for a multi-dimensional problem, are given. The main advantage of the scheme proposed here compared with earlier methods is that the scheme is linear and scalable to large problems.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.12.002
       
  • Data assimilation using noisy time-averaged measurements
    • Authors: Jordan Blocher; Vincent R. Martinez; Eric Olson
      Abstract: Publication date: Available online 8 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Jordan Blocher, Vincent R. Martinez, Eric Olson
      We study the synchronization of chaotic systems when the coupling between them contains both time averages and stochastic noise. Our model dynamics—inspired by the partial differential equations which govern the atmosphere—are given by the Lorenz equations which are a system of three ordinary differential equations in the variables X , Y and Z . Our theoretical results show that coupling two copies of the Lorenz equations using a feedback control which consists of time averages of the X variable leads to exact synchronization provided the time-averaging window is known and sufficiently small. In the presence of noise the convergence is to within a factor of the variance of the noise. We also consider the case when the time-averaging window is not known and show that it is possible to tune the feedback control to recover the size of the time-averaging window. Further numerical computations show that synchronization is more accurate and occurs under much less stringent conditions than our theory requires.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.12.004
       
  • Turing patterns in parabolic systems of conservation laws and numerically
           observed stability of periodic waves
    • Authors: Blake Barker; Soyeun Jung; Kevin Zumbrun
      Abstract: Publication date: Available online 7 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Blake Barker, Soyeun Jung, Kevin Zumbrun
      Turing patterns on unbounded domains have been widely studied in systems of reaction–diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from subcritical Turing bifurcations. This answers in the affirmative a question of Oh-Zumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full small-amplitude stability diagram–specifically, determination of rigorous Eckhaus-type stability conditions–remains an interesting open problem.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.12.003
       
  • Existence of an attractor and determining modes for structurally damped
           nonlinear wave equations
    • Authors: B.A. Bilgin; V.K. Kalantarov
      Abstract: Publication date: Available online 6 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): B.A. Bilgin, V.K. Kalantarov
      The paper is devoted to the study of asymptotic behavior as t → + ∞ of solutions of initial boundary value problem for structurally damped semi-linear wave equation ∂ t 2 u ( x , t ) − Δ u ( x , t ) + γ ( − Δ ) θ ∂ t u ( x , t ) + f ( u ) = g ( x ) , θ ∈ ( 0 , 1 ) , x ∈ Ω , t > 0 under homogeneous Dirichlet’s boundary condition in a bounded domain Ω ⊂ R 3 . We proved that the asymptotic behavior as t → ∞ of solutions of this problem is completely determined by dynamics of the first N Fourier modes, when N is large enough. We also proved that the semigroup generated by this problem when θ ∈ ( 1 2 , 1 ) possesses an exponential attractor.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.12.001
       
  • The Maxwell–Boltzmann approximation for ion kinetic modeling
    • Authors: Claude Bardos; François Golse; Toan T. Nguyen; Rémi Sentis
      Abstract: Publication date: Available online 5 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Claude Bardos, François Golse, Toan T. Nguyen, Rémi Sentis
      The aim of this paper is to provide a justification of the Maxwell–Boltzmann approximation of electron density from kinetic models. First, under reasonable regularity assumption, we rigorously derive a reduced kinetic model for the dynamics of ions, while electrons satisfy the Maxwell–Boltzmann relation. Second, we prove that equilibria of the electrons distribution are local Maxwellians, and they can be uniquely determined from conserved mass and energy constants. Finally, we prove that the reduced kinetic model for ions is globally well-posed. The constructed weak solutions conserve energy.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.10.014
       
  • On the connection between Kolmogorov microscales and friction in pipe
           flows of viscoplastic fluids
    • Authors: H.R. Anbarlooei; D.O.A. Cruz; F. Ramos; C.M.M. Santos; A.P. Silva Freire
      Abstract: Publication date: Available online 5 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): H.R. Anbarlooei, D.O.A. Cruz, F. Ramos, C.M.M. Santos, A.P. Silva Freire
      The present work extends Kolmogorov’s micro-scales to a large family of viscoplastic fluids. The new micro-scales, combined with Gioia and Chakaborty’s (2006) friction phenomenology theory, lead to a unified framework for the description of the friction coefficient in turbulent flows. A resulting Blasius-type friction equation is tested against some available experimental data and shows good agreement over a significant range of Hedstrom and Reynolds numbers. The work also comments on the role of the new expression as a possible benchmark test for the convergence of DNS simulations. The formula also provides limits for the maximum drag reduction of viscoplastic flows.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.005
       
  • Travelling waves and their bifurcations in the Lorenz-96 model
    • Authors: Dirk L. van Kekem; Alef E. Sterk
      Abstract: Publication date: Available online 5 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Dirk L. van Kekem, Alef E. Sterk
      In this paper we study the dynamics of the monoscale Lorenz-96 model using both analytical and numerical means. The bifurcations for positive forcing parameter F are investigated. The main analytical result is the existence of Hopf or Hopf-Hopf bifurcations in any dimension n ≥ 4 . Exploiting the circulant structure of the Jacobian matrix enables us to reduce the first Lyapunov coefficient to an explicit formula from which it can be determined when the Hopf bifurcation is sub- or supercritical. The first Hopf bifurcation for F > 0 is always supercritical and the periodic orbit born at this bifurcation has the physical interpretation of a travelling wave. Furthermore, by unfolding the codimension two Hopf-Hopf bifurcation it is shown to act as an organising centre, explaining dynamics such as quasi-periodic attractors and multistability, which are observed in the original Lorenz-96 model. Finally, the region of parameter values beyond the first Hopf bifurcation value is investigated numerically and routes to chaos are described using bifurcation diagrams and Lyapunov exponents. The observed routes to chaos are various but without clear pattern as n → ∞ .

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.008
       
  • Gradual multifractal reconstruction of time-series: Formulation of the
           method and an application to the coupling between stock market indices and
           their Hölder exponents
    • Authors: Christopher J. Keylock
      Abstract: Publication date: Available online 5 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Christopher J. Keylock
      A technique termed gradual multifractal reconstruction (GMR) is formulated. A continuum is defined from a signal that preserves the pointwise Hölder exponent (multifractal) structure of a signal but randomises the locations of the original data values with respect to this ( φ = 0 ), to the original signal itself ( φ = 1 ). We demonstrate that this continuum may be populated with synthetic time series by undertaking selective randomisation of wavelet phases using a dual-tree complex wavelet transform. That is, the φ = 0 end of the continuum is realised using the recently proposed iterated, amplitude adjusted wavelet transform algorithm (Keylock, 2017) that fully randomises the wavelet phases. This is extended to the GMR formulation by selective phase randomisation depending on whether or not the wavelet coefficient amplitudes exceeds a threshold criterion. An econophysics application of the technique is presented. The relation between the normalised log-returns and their Hölder exponents for the daily returns of eight financial indices are compared. One particularly noticeable result is the change for the two american indices (NASDAQ 100 and S&P 500) from a non-significant to a strongly significant (as determined using GMR) cross-correlation between the returns and their Hölder exponents from before the 2008 crash to afterwards. This is also reflected in the skewness of the phase difference distributions, which exhibit a geographical structure, with asian markets not exhibiting significant skewness in contrast to those from elsewhere globally.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.011
       
  • Determining modes for the 3D Navier–Stokes equations
    • Authors: Alexey Cheskidov; Mimi Dai; Landon Kavlie
      Abstract: Publication date: Available online 5 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alexey Cheskidov, Mimi Dai, Landon Kavlie
      We introduce a determining wavenumber for the forced 3D Navier–Stokes equations (NSE) defined for each individual solution. Even though this wavenumber blows up if the solution blows up, its time average is uniformly bounded for all solutions on the weak global attractor. The bound is compared to Kolmogorov’s dissipation wavenumber and the Grashof constant.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.014
       
  • New integrable model of propagation of the few-cycle pulses in an
           anisotropic microdispersed medium
    • Authors: S.V. Sazonov; N.V. Ustinov
      Abstract: Publication date: Available online 2 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): S.V. Sazonov, N.V. Ustinov
      We investigate the propagation of the few-cycle electromagnetic pulses in the anisotropic microdispersed medium. The effects of the anisotropy and spatial dispersion of the medium are created by the two sorts of the two-level atoms. The system of the material equations describing an evolution of the states of the atoms and the wave equations for the ordinary and extraordinary components of the pulses is derived. By applying the approximation of the sudden excitation to exclude the material variables, we reduce this system to the single nonlinear wave equation that generalizes the modified sine–Gordon equation and the Rabelo–Fokas equation. It is shown that this equation is integrable by means of the inverse scattering transformation method if an additional restriction on the parameters is imposed. The multisoliton solutions of this integrable generalization are constructed and investigated.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.012
       
  • Initiation of reaction–diffusion waves of blood coagulation
    • Authors: Galochkina Marion; Volpert
      Abstract: Publication date: Available online 2 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): T. Galochkina, M. Marion, V. Volpert
      Formation of blood clot in response to the vessel damage is triggered by the complex network of biochemical reactions of the coagulation cascade. The process of clot growth can be modeled as a traveling wave solution of the bistable reaction–diffusion system. The critical value of the initial condition which leads to convergence of the solution to the traveling wave corresponds to the pulse solution of the corresponding stationary problem. In the current study we prove the existence of the pulse solution for the stationary problem in the model of the main reactions of the blood coagulation cascade using the Leray–Schauder method.

      PubDate: 2017-12-12T12:38:45Z
       
  • Decay of Kadomtsev–Petviashvili lumps in dissipative media
    • Authors: S. Clarke; K. Gorshkov; R. Grimshaw; Y. Stepanyants
      Abstract: Publication date: Available online 2 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): S. Clarke, K. Gorshkov, R. Grimshaw, Y. Stepanyants
      The decay of Kadomtsev–Petviashvili lumps is considered for a few typical dissipations –Rayleigh dissipation, Reynolds dissipation, Landau damping, Chezy bottom friction, viscous dissipation in the laminar boundary layer, and radiative losses caused by large-scale dispersion. It is shown that the straight-line motion of lumps is unstable under the influence of dissipation. The lump trajectories are calculated for two most typical models of dissipation –the Rayleigh and Reynolds dissipations. A comparison of analytical results obtained within the framework of asymptotic theory with the direct numerical calculations of the Kadomtsev–Petviashvili equation is presented. Good agreement between the theoretical and numerical results is obtained.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.009
       
  • Existence time for the 3D Navier–Stokes equations in a generalized
           Gevrey class
    • Authors: Animikh Biswas; Ciprian Foias; Basil Nicolaenko
      Abstract: Publication date: Available online 2 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Animikh Biswas, Ciprian Foias, Basil Nicolaenko
      Gevrey class technique is a widely used tool for studying higher regularity properties of solutions to dissipative equations. Maximal radius in a Gevrey class determines a small length scale associated to the decay of the Fourier power spectrum and turbulence. In this paper, we consider existence theory for the three dimensional, incompressible Navier–Stokes equations, in a certain generalized Gevrey class of functions, which contains all the analytic and sub-analytic Gevrey classes, and is in turn contained in the space of all C ∞ functions. This class has been in focus recently, pertaining to the study of the attractor for the 2D Navier–Stokes equations, particularly in relation to a question posed by Peter Constantin as to whether or not zero is in the attractor of the 2D Navier–Stokes equations. We show that in the 3D case, the differential inequality that one obtains in this class is almost linear, and the corresponding existence time is better than the reciprocal of any algebraic power of the norm of the initial data (in this class). Subsequently, we compare the existence times with well-known ones in Sobolev classes for certain types of initial data. We also obtain a lower bound for the maximal radius in the generalized Gevrey class under consideration here. The first two authors are delighted to dedicate this paper to Professor Edriss Titi on the occasion of his 60th birthday. They much admire Professor Titi’s enormous research contributions to the field of fluid dynamics, as well as his mathematical acumen, novel ideas, and infectious enthusiasm for the subject, as evidenced by his numerous lucid, and enthralling, lectures. This research was completed after the passing of Professor Basil Nicolaenko. The first two authors are confident that he too would have shared their sentiments concerning Professor Titi.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.013
       
  • Interaction of non-radially symmetric camphor particles
    • Authors: Shin-Ichiro Ei; Hiroyuki Kitahata; Yuki Koyano; Masaharu Nagayama
      Abstract: Publication date: Available online 2 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Shin-Ichiro Ei, Hiroyuki Kitahata, Yuki Koyano, Masaharu Nagayama
      In this study, the interaction between two non-radially symmetric camphor particles is theoretically investigated and the equation describing the motion is derived as an ordinary differential system for the locations and the rotations. In particular, slightly modified non-radially symmetric cases from radial symmetry are extensively investigated and explicit motions are obtained. For example, it is theoretically shown that elliptically deformed camphor particles interact so as to be parallel with major axes. Such predicted motions are also checked by real experiments and numerical simulations.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.004
       
  • Modeling ultrashort electromagnetic pulses with a generalized
           Kadomtsev–Petviashvili equation
    • Authors: A. Hofstrand; J.V. Moloney
      Abstract: Publication date: Available online 2 December 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): A. Hofstrand, J.V. Moloney
      In this paper we derive a properly scaled model for the nonlinear propagation of intense, ultrashort, mid-infrared electromagnetic pulses (10-100 femtoseconds) through an arbitrary dispersive medium. The derivation results in a generalized Kadomtsev–Petviashvili (gKP) equation. In contrast to envelope-based models such as the Nonlinear Schrödinger (NLS) equation, the gKP equation describes the dynamics of the field’s actual carrier wave. It is important to resolve these dynamics when modeling ultrashort pulses. We proceed by giving an orginal proof of sufficient conditions on the initial pulse for a singularity to form in the field after a finite propagation distance. The model is then numerically simulated in 2D using a spectral-solver with initial data and physical parameters highlighting our theoretical results.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.010
       
  • The role of BKM-type theorems in 3D Euler, Navier–Stokes and
           Cahn-Hilliard-Navier–Stokes analysis
    • Authors: John D. Gibbon; Anupam Gupta; Nairita Pal; Rahul Pandit
      Abstract: Publication date: Available online 23 November 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): John D. Gibbon, Anupam Gupta, Nairita Pal, Rahul Pandit
      The Beale-Kato-Majda theorem contains a single criterion that controls the behaviour of solutions of the 3 D incompressible Euler equations. Versions of this theorem are discussed in terms of the regularity issues surrounding the 3 D incompressible Euler and Navier–Stokes equations together with a phase-field model for the statistical mechanics of binary mixtures called the 3 D Cahn-Hilliard-Navier–Stokes (CHNS) equations. A theorem of BKM-type is established for the CHNS equations for the full parameter range. Moreover, for this latter set, it is shown that there exists a Reynolds number and a bound on the energy-dissipation rate that, remarkably, reproduces the R e 3 ∕ 4 upper bound on the inverse Kolmogorov length normally associated with the Navier–Stokes equations alone. An alternative length-scale is introduced and discussed, together with a set of pseudo-spectral computations on a 12 8 3 grid.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.007
       
  • Wave propagation in a strongly nonlinear locally resonant granular crystal
    • Authors: K. Vorotnikov; Y. Starosvetsky; G. Theocharis; P.G. Kevrekidis
      Abstract: Publication date: Available online 23 November 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): K. Vorotnikov, Y. Starosvetsky, G. Theocharis, P.G. Kevrekidis
      In this work, we study the wave propagation in a recently proposed acoustic structure, the locally resonant granular crystal. This structure is composed of a one-dimensional granular crystal of hollow spherical particles in contact, containing linear resonators. The relevant model is presented and examined through a combination of analytical approximations (based on ODE and nonlinear map analysis) and of numerical results. The generic dynamics of the system involves a degradation of the well-known traveling pulse of the standard Hertzian chain of elastic beads. Nevertheless, the present system is richer, in that as the primary pulse decays, secondary ones emerge and eventually interfere with it creating modulated wavetrains. Remarkably, upon suitable choices of parameters, this interference “distills” a weakly nonlocal solitary wave (a “nanopteron”). This motivates the consideration of such nonlinear structures through a separate Fourier space technique, whose results suggest the existence of such entities not only with a single-side tail, but also with periodic tails on both ends. These tails are found to oscillate with the intrinsic oscillation frequency of the out-of-phase motion between the outer hollow bead and its internal linear attachment.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.10.007
       
  • Global dynamics for switching systems and their extensions by linear
           differential equations
    • Authors: Zane Huttinga; Bree Cummins; Tomáš Gedeon; Konstantin Mischaikow
      Abstract: Publication date: Available online 15 November 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Zane Huttinga, Bree Cummins, Tomáš Gedeon, Konstantin Mischaikow
      Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.003
       
  • On the manifestation of coexisting nontrivial equilibria leading to
           potential well escapes in an inhomogeneous floating body
    • Authors: Dane Sequeira; Xue-She Wang; B.P. Mann
      Abstract: Publication date: Available online 14 November 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Dane Sequeira, Xue-She Wang, B.P. Mann
      This paper examines the bifurcation and stability behavior of inhomogeneous floating bodies, specifically a rectangular prism with asymmetric mass distribution. A nonlinear model is developed to determine the stability of the upright and tilted equilibrium positions as a function of the vertical position of the center of mass within the prism. These equilibria positions are defined by an angle of rotation and a vertical position where rotational motion is restricted to a two dimensional plane. Numerical investigations are conducted using path-following continuation methods to determine equilibria solutions and evaluate stability. Bifurcation diagrams and basins of attraction that illustrate the stability of the equilibrium positions as a function of the vertical position of the center of mass within the prism are generated. These results reveal complex stability behavior with many coexisting solutions. Static experiments are conducted to validate equilibria orientations against numerical predictions with results showing good agreement. Dynamic experiments that examine potential well hopping behavior in a waveflume for various wave conditions are also conducted.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.002
       
  • Three-wave resonant interactions: Multi-dark-dark-dark solitons,
           breathers, rogue waves, and their interactions and dynamics
    • Authors: Guoqiang Zhang; Zhenya Yan; Xiao-Yong Wen
      Abstract: Publication date: Available online 10 November 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Guoqiang Zhang, Zhenya Yan, Xiao-Yong Wen
      We investigate three-wave resonant interactions through both the generalized Darboux transformation method and numerical simulations. Firstly, we derive a simple multi-dark-dark-dark-soliton formula through the generalized Darboux transformation. Secondly, we use the matrix analysis method to avoid the singularity of transformed potential functions and to find the general nonsingular breather solutions. Moreover, through a limit process, we deduce the general rogue wave solutions and give a classification by their dynamics including bright, dark, four-petals, and two-peaks rogue waves. Ever since the coexistence of dark soliton and rogue wave in non-zero background, their interactions naturally become a quite appealing topic. Based on the N -fold Darboux transformation, we can derive the explicit solutions to depict their interactions. Finally, by performing extensive numerical simulations we can predict whether these dark solitons and rogue waves are stable enough to propagate. These results can be available for several physical subjects such as fluid dynamics, nonlinear optics, solid state physics, and plasma physics.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.11.001
       
  • Oscillating solutions of the Vlasov–Poisson system—A numerical
           investigation
    • Authors: Tobias Ramming; Gerhard Rein
      Abstract: Publication date: Available online 6 November 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Tobias Ramming, Gerhard Rein
      Numerical evidence is given that spherically symmetric perturbations of stable spherically symmetric steady states of the gravitational Vlasov–Poisson system lead to solutions which oscillate in time. The oscillations can be periodic in time or damped. Along one-parameter families of polytropic steady states we establish an Eddington–Ritter type relation which relates the period of the oscillation to the central density of the steady state. The numerically obtained periods are used to estimate possible periods for typical elliptical galaxies.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.10.013
       
  • Well posedness and maximum entropy approximation for the dynamics of
           quantitative traits
    • Authors: Katarína Boďová; Jan Haskovec; Peter Markowich
      Abstract: Publication date: Available online 6 November 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Katarína Boďová, Jan Haskovec, Peter Markowich
      We study the Fokker–Planck equation derived in the large system limit of the Markovian process describing the dynamics of quantitative traits. The Fokker–Planck equation is posed on a bounded domain and its transport and diffusion coefficients vanish on the domain’s boundary. We first argue that, despite this degeneracy, the standard no-flux boundary condition is valid. We derive the weak formulation of the problem and prove the existence and uniqueness of its solutions by constructing the corresponding contraction semigroup on a suitable function space. Then, we prove that for the parameter regime with high enough mutation rate the problem exhibits a positive spectral gap, which implies exponential convergence to equilibrium. Next, we provide a simple derivation of the so-called Dynamic Maximum Entropy (DynMaxEnt) method for approximation of observables (moments) of the Fokker–Planck solution, which can be interpreted as a nonlinear Galerkin approximation. The limited applicability of the DynMaxEnt method inspires us to introduce its modified version that is valid for the whole range of admissible parameters. Finally, we present several numerical experiments to demonstrate the performance of both the original and modified DynMaxEnt methods. We observe that in the parameter regimes where both methods are valid, the modified one exhibits slightly better approximation properties compared to the original one.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.10.015
       
  • Patterns and coherence resonance in the stochastic Swift–Hohenberg
           equation with Pyragas control: The Turing bifurcation case
    • Authors: R. Kuske; C.Y. Lee; V. Rottschäfer
      Abstract: Publication date: Available online 2 November 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): R. Kuske, C.Y. Lee, V. Rottschäfer
      We provide a multiple time scales analysis for the Swift–Hohenberg equation with delayed feedback via Pyragas control, with and without additive noise. An analysis of the pattern formation near onset indicates both the possibility of either standing waves (rolls) or traveling waves via Turing or Turing–Hopf bifurcations, respectively, depending on the product of the strength of the feedback and the length of the delay. The remainder of the paper is focused on Turing bifurcations, where the delay can drive the appearance of an additional time scale, intermediate to the usual slow and fast time scales observed in the modulation of rolls without delay. In the deterministic case, a Ginzburg–Landau-type modulation equation is derived that inherits Pyragas control terms from the original equation. The Eckhaus stability criteria is obtained for the rolls, with the intermediate time scale observed in the transients. In the stochastic context, slow modulation equations are derived for the amplitudes of the primary modes that are coupled to a fast Ornstein–Uhlenbeck-type equation with delay for the zero mode driven by the additive noise. By deriving an averaging approximation for the amplitude of the primary mode, we show how the interaction of noise and delay influences the existence and stability range for the noisy roll-type patterns. Furthermore, approximations for the spectral densities of the primary and zero modes show that oscillations on the intermediate times scale are sustained through the phenomenon of coherence resonance. These dynamics on the intermediate time scale are sustained through the interaction of noise and delay, in contrast to the deterministic context where dynamics on the intermediate times scale are transient.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.10.012
       
  • On the Lagrangian and Eulerian analyticity for the Euler equations
    • Authors: Guher Camliyurt; Igor Kukavica
      Abstract: Publication date: Available online 1 November 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Guher Camliyurt, Igor Kukavica
      We address preservation of the Lagrangian analyticity radius of solutions to the Euler equations belonging to natural analytic space based on the size of Taylor (or Gevrey) coefficients. We prove that if the solution belongs to such space, then the solution also belongs to it for a positive amount of time. We also prove the local analog of this result for a sufficiently large Gevrey parameter; however, we show that the preservation holds independently of the size of the radius. Finally, we construct a solution which shows that the first result does not hold in the Eulerian setting.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.09.006
       
  • Homogenization of one-dimensional draining through heterogeneous porous
           media including higher-order approximations
    • Authors: Daniel M. Anderson; Richard M. McLaughlin; Cass T. Miller
      Abstract: Publication date: Available online 31 October 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Daniel M. Anderson, Richard M. McLaughlin, Cass T. Miller
      We examine a mathematical model of one-dimensional draining of a fluid through a periodically-layered porous medium. A porous medium, initially saturated with a fluid of a high density is assumed to drain out the bottom of the porous medium with a second lighter fluid replacing the draining fluid. We assume that the draining layer is sufficiently dense that the dynamics of the lighter fluid can be neglected with respect to the dynamics of the heavier draining fluid and that the height of the draining fluid, represented as a free boundary in the model, evolves in time. In this context, we neglect interfacial tension effects at the boundary between the two fluids. We show that this problem admits an exact solution. Our primary objective is to develop a homogenization theory in which we find not only leading-order, or effective, trends but also capture higher-order corrections to these effective draining rates. The approximate solution obtained by this homogenization theory is compared to the exact solution for two cases: (1) the permeability of the porous medium varies smoothly but rapidly and (2) the permeability varies as a piecewise constant function representing discrete layers of alternating high/low permeability. In both cases we are able to show that the corrections in the homogenization theory accurately predict the position of the free boundary moving through the porous medium.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.10.010
       
  • Statistical solutions and Onsager’s conjecture
    • Authors: U.S. Fjordholm; E. Wiedemann
      Abstract: Publication date: Available online 28 October 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): U.S. Fjordholm, E. Wiedemann
      We prove a version of Onsager’s conjecture on the conservation of energy for the incompressible Euler equations in the context of statistical solutions, as introduced recently by Fjordholm et al. (2017). As a byproduct, we also obtain an alternative proof for the conservative direction of Onsager’s conjecture for weak solutions, under a weaker Besov-type regularity assumption than previously known.

      PubDate: 2017-12-12T12:38:45Z
      DOI: 10.1016/j.physd.2017.10.009
       
  • Energy transfer in autoresonant Klein-Gordon chains
    • Authors: Agnessa Kovaleva
      Abstract: Publication date: Available online 12 October 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Agnessa Kovaleva
      In this work we examine autoresonant oscillations in a Klein-Gordon chain of finite length. The chain is subjected to an external periodic forcing with a slowly varying frequency applied at one edge of the chain. Explicit asymptotic equations describing the amplitudes and the phases of oscillations are derived. These equations demonstrate that, in contrast to the chains with linear attachments, the nonlinear chain can be entirely captured into resonance provided that its structural and excitation parameters exceed their critical thresholds. It is shown that at large times the amplitudes of AR oscillations converge to a monotonically growing mean amplitude that is equal for all oscillators. The threshold values of the structural and excitation parameters, which allow the emergence of autoresonance in the entire chain, are determined. The derived analytic results are in good agreement with numerical simulation.

      PubDate: 2017-10-13T18:11:50Z
      DOI: 10.1016/j.physd.2017.10.003
       
  • Identification of particle-laden flow features from wavelet decomposition
    • Authors: A. Jackson; B. Turnbull
      Abstract: Publication date: Available online 10 October 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): A. Jackson, B. Turnbull
      A wavelet decomposition based technique is applied to air pressure data obtained from laboratory-scale powder snow avalanches. This technique is shown to be a powerful tool for identifying both repeatable and chaotic features at any frequency within the signal. Additionally, this technique is demonstrated to be a robust method for the removal of noise from the signal as well as being capable of removing other contaminants from the signal. Whilst powder snow avalanches are the focus of the experiments analysed here, the features identified can provide insight to other particle-laden gravity currents and the technique described is applicable to a wide variety of experimental signals.

      PubDate: 2017-10-11T18:08:12Z
      DOI: 10.1016/j.physd.2017.09.009
       
  • Semi-global persistence and stability for a class of forced discrete-time
           population models
    • Authors: Daniel Franco; Chris Guiver; Hartmut Logemann; Juan Perán
      Abstract: Publication date: Available online 31 August 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Daniel Franco, Chris Guiver, Hartmut Logemann, Juan Perán
      We consider persistence and stability properties for a class of forced discrete-time difference equations with three defining properties: the solution is constrained to evolve in the non-negative orthant, the forcing acts multiplicatively, and the dynamics are described by so-called Lur’e systems, containing both linear and non-linear terms. Many discrete-time biological models encountered in the literature may be expressed in the form of a Lur’e system and, in this context, the multiplicative forcing may correspond to harvesting, culling or time-varying (such as seasonal) vital rates or environmental conditions. Drawing upon techniques from systems and control theory, and assuming that the forcing is bounded, we provide conditions under which persistence occurs and, further, that a unique non-zero equilibrium is stable with respect to the forcing in a sense which is reminiscent of input-to-state stability, a concept well-known in nonlinear control theory. The theoretical results are illustrated with several examples. In particular, we discuss how our results relate to previous literature on stabilization of chaotic systems by so-called proportional feedback control.

      PubDate: 2017-09-26T17:33:44Z
      DOI: 10.1016/j.physd.2017.08.001
       
  • Operator splitting method for simulation of dynamic flows in natural gas
           pipeline networks
    • Authors: Sergey A. Dyachenko; Anatoly Zlotnik; Alexander O. Korotkevich; Michael Chertkov
      Abstract: Publication date: Available online 19 September 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Sergey A. Dyachenko, Anatoly Zlotnik, Alexander O. Korotkevich, Michael Chertkov
      We develop an operator splitting method to simulate flows of isothermal compressible natural gas over transmission pipelines. The method solves a system of nonlinear hyperbolic partial differential equations (PDEs) of hydrodynamic type for mass flow and pressure on a metric graph, where turbulent losses of momentum are modeled by phenomenological Darcy-Weisbach friction. Mass flow balance is maintained through the boundary conditions at the network nodes, where natural gas is injected or withdrawn from the system. Gas flow through the network is controlled by compressors boosting pressure at the inlet of the adjoint pipe. Our operator splitting numerical scheme is unconditionally stable and it is second order accurate in space and time. The scheme is explicit, and it is formulated to work with general networks with loops. We test the scheme over range of regimes and network configurations, also comparing its performance with performance of two other state of the art implicit schemes.

      PubDate: 2017-09-20T17:10:15Z
      DOI: 10.1016/j.physd.2017.09.002
       
  • Phase models and clustering in networks of oscillators with delayed
           coupling
    • Authors: Sue Ann Campbell; Zhen Wang
      Abstract: Publication date: Available online 19 September 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Sue Ann Campbell, Zhen Wang
      We consider a general model for a network of oscillators with time delayed coupling where the coupling matrix is circulant. We use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to determine model independent existence and stability results for symmetric cluster solutions. Our results extend previous work to systems with time delay and a more general coupling matrix. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We apply our analytical results to a network of Morris Lecar neurons and compare these results with numerical continuation and simulation studies.

      PubDate: 2017-09-20T17:10:15Z
      DOI: 10.1016/j.physd.2017.09.004
       
  • 4-wave dynamics in kinetic wave turbulence
    • Authors: Sergio Chibbaro; Giovanni Dematteis; Lamberto Rondoni
      Abstract: Publication date: Available online 18 September 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Sergio Chibbaro, Giovanni Dematteis, Lamberto Rondoni
      A general Hamiltonian wave system with quartic resonances is considered, in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. The evolution equation for the multimode characteristic function Z is obtained within an “interaction representation” and a perturbation expansion in the small nonlinearity parameter. A frequency renormalization is performed to remove linear terms that do not appear in the 3-wave case. Feynman-Wyld diagrams are used to average over phases, leading to a first order differential evolution equation for Z . A hierarchy of equations, analogous to the Boltzmann hierarchy for low density gases is derived, which preserves in time the property of random phases and amplitudes. This amounts to a general formalism for both the N -mode and the 1-mode PDF equations for 4-wave turbulent systems, suitable for numerical simulations and for investigating intermittency. Some of the main results which are developed here in details have been tested numerically in a recent work.

      PubDate: 2017-09-20T17:10:15Z
      DOI: 10.1016/j.physd.2017.09.001
       
  • Agent-based model of the effect of globalization on inequality and class
           mobility
    • Authors: Joep H.M. Evers; David Iron; Theodore Kolokolnikov; John Rumsey
      Abstract: Publication date: Available online 14 September 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Joep H.M. Evers, David Iron, Theodore Kolokolnikov, John Rumsey
      We consider a variant of the Bouchaud-Mézard model for wealth distribution in a society which incorporates the interaction radius between the agents, to model the extent of globalization in a society. The wealth distribution depends critically on the extent of this interaction. When interaction is relatively local, a small cluster of individuals emerges which accumulate most of the society’s wealth. In this regime, the society is highly stratified with little or no class mobility. As the interaction is increased, the number of wealthy agents decreases, but the overall inequality rises as the freed-up wealth is transferred to the remaining wealthy agents. However when the interaction exceeds a certain critical threshold, the society becomes highly mobile resulting in a much lower economic inequality (low Gini index). This is consistent with the Kuznets upside-down U shaped inequality curve hypothesis.

      PubDate: 2017-09-20T17:10:15Z
      DOI: 10.1016/j.physd.2017.08.009
       
  • The stability spectrum for elliptic solutions to the sine-Gordon equation
    • Authors: Bernard Deconinck; Peter McGill; Benjamin L. Segal
      Abstract: Publication date: Available online 14 September 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Bernard Deconinck, Peter McGill, Benjamin L. Segal
      We present an analysis of the stability spectrum for all stationary periodic solutions to the sine-Gordon equation. An analytical expression for the spectrum is given. From this expression, various quantitative and qualitative results about the spectrum are derived. Specifically, the solution parameter space is shown to be split into regions of distinct qualitative behavior of the spectrum, in one of which the solutions are stable. Additional results on the spectral stability of solutions with respect to perturbations of an integer multiple of the solution period are given.

      PubDate: 2017-09-14T16:57:02Z
      DOI: 10.1016/j.physd.2017.08.010
       
  • Multiple equilibria, bifurcations and selection scenarios in cosymmetric
           problem of thermal convection in porous medium
    • Authors: Vasily N. Govorukhin; Igor V. Shevchenko
      Abstract: Publication date: Available online 14 September 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Vasily N. Govorukhin, Igor V. Shevchenko
      We study convection in a two-dimensional container of porous material saturated with fluid and heated from below. This problem belongs to the class of dynamical systems with nontrivial cosymmetry. The cosymmetry gives rise to a hidden parameter in the system and continuous families of infinitely many equilibria, and leads to non-trivial bifurcations. In this article we present our numerical studies that demonstrate nonlinear phenomena resulting from the existence of cosymmetry. We give a comprehensive picture of different bifurcations which occur in cosymmetric dynamical systems and in the convection problem. It includes internal and external (as an invariant set) bifurcations of one-parameter families of equilibria, as well as bifurcations leading to periodic, quasiperiodic and chaotic behaviour. The existence of infinite number of stable steady-state regimes begs the important question as to which of them can realize in physical experiments. In the paper, this question (known as the selection problem) is studied in detail. In particular, we show that the selection scenarios strongly depend on the initial temperature distribution of the fluid. The calculations are carried out by the global cosymmetry-preserving Galerkin method, and numerical methods used to analyse cosymmetric systems are also described.

      PubDate: 2017-09-14T16:57:02Z
      DOI: 10.1016/j.physd.2017.08.012
       
  • Bounded ultra-elliptic solutions of the defocusing nonlinear
           Schrödinger equation
    • Authors: Otis C. Wright; III
      Abstract: Publication date: Available online 12 September 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Otis C. Wright III
      An effective integration method is presented for the bounded ultra-elliptic solutions of the defocusing nonlinear Schrödinger equation. The two-phase solutions are explicitly parametrized in terms of two physically-meaningful variables: the energy density and the momentum density. Cavitation, viz., a minimum amplitude of zero, occurs if and only if the length of the largest spectral band is less than or equal to the sum of the lengths of the two smaller spectral bands. In the case of strict inequality, there are exactly two cavitation points in each period parallelogram.

      PubDate: 2017-09-14T16:57:02Z
      DOI: 10.1016/j.physd.2017.08.013
       
  • A coherent structure approach for parameter estimation in Lagrangian Data
           Assimilation
    • Authors: John Maclean; Naratip Santitissadeekorn; Christopher K.R.T. Jones
      Abstract: Publication date: Available online 5 September 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): John Maclean, Naratip Santitissadeekorn, Christopher K.R.T. Jones
      We introduce a data assimilation method to estimate model parameters with observations of passive tracers by directly assimilating Lagrangian Coherent Structures. Our approach differs from the usual Lagrangian Data Assimilation approach, where parameters are estimated based on tracer trajectories. We employ the Approximate Bayesian Computation (ABC) framework to avoid computing the likelihood function of the coherent structure, which is usually unavailable. We solve the ABC by a Sequential Monte Carlo (SMC) method, and use Principal Component Analysis (PCA) to identify the coherent patterns from tracer trajectory data. Our new method shows remarkably improved results compared to the bootstrap particle filter when the physical model exhibits chaotic advection.

      PubDate: 2017-09-09T06:52:08Z
      DOI: 10.1016/j.physd.2017.08.007
       
  • A classical limit-cycle system that mimics the quantum-mechanical harmonic
           oscillator
    • Authors: Yair Zarmi
      Abstract: Publication date: Available online 31 August 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Yair Zarmi
      Classical harmonic oscillators affected by appropriately chosen nonlinear dissipative perturbations can exhibit infinite sequences of limit cycles, which mimic quantized systems. For properly chosen perturbations, the large-amplitude limit cycles approach circles. The higher the amplitude of the limit cycle is, the smaller are the dissipative deviations from energy conservation. The weaker the perturbation is, the earlier on does the asymptotic behavior show up already in low-lying limit cycles. Simple modifications of the Rayleigh and van der Pol oscillators yield infinite sequences of limit cycles such that the energy spectrum of the higher-amplitude limit cycles tends to that of the quantum-mechanical particle in a box. For another judiciously chosen dissipative perturbation, the energy spectrum of the higher-amplitude limit cycles tends to that of the quantum-mechanical harmonic oscillator. In all cases, one first finds the limit-cycle solutions for dissipation strength, ε ≠ 0 . The “energy of each limit cycle” then oscillates around an average value. In the limit ε → 0 these oscillations vanish, and the limit cycles in the infinite sequence attain constant values for their energies, a characteristic that is required for such classical systems to mimic Hamiltonian quantum-mechanical systems.

      PubDate: 2017-09-02T22:27:45Z
      DOI: 10.1016/j.physd.2017.08.003
       
  • Dynamical systems analysis of the Maasch–Saltzman model for glacial
           cycles
    • Authors: Hans Engler; Hans G. Kaper; Tasso J. Kaper; Theodore Vo
      Abstract: Publication date: Available online 24 August 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Hans Engler, Hans G. Kaper, Tasso J. Kaper, Theodore Vo
      This article is concerned with the internal dynamics of a conceptual model proposed by Maasch and Saltzman (1990) to explain central features of the glacial cycles observed in the climate record of the Pleistocene Epoch. It is shown that, in most parameter regimes, the long-term system dynamics occur on certain intrinsic two-dimensional invariant manifolds in the three-dimensional state space. These invariant manifolds are slow manifolds when the characteristic time scales for the total global ice mass and the strength of the North Atlantic Deep Water circulation are well-separated, and they are center manifolds when these characteristic time scales are comparable. In both cases, the reduced dynamics on these manifolds are governed by Bogdanov-Takens singularities, and the bifurcation curves associated to these singularities organize the parameter regions in which the model exhibits glacial cycles. In addition, knowledge of the reduced systems and their bifurcations is useful for understanding the effects slowly varying parameters, which cause passage through Hopf bifurcations, and of orbital (Milankovitch) forcing. Both are central to the mechanism proposed by Maasch and Saltzman for the mid-Pleistocene transition in their model.

      PubDate: 2017-09-02T22:27:45Z
      DOI: 10.1016/j.physd.2017.08.006
       
  • Modeling and numerical investigations for hierarchical pattern formation
           in desiccation cracking
    • Authors: Sayako Hirobe; Kenji Oguni
      Abstract: Publication date: Available online 16 August 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Sayako Hirobe, Kenji Oguni
      Desiccation cracking and its pattern formation are widely observed in nature. The network of the surface cracks forms polygonal cells with typical size. This crack pattern is not formed in a simultaneous manner, instead, formed in a sequential and hierarchical manner. The strain energy accumulated by the heterogeneous drying shrinkage strain is systematically released by the cracks. In this sense, desiccation cracking phenomenon can be regarded as a typical example of the pattern formation in the dynamical system with dissipation. We propose a mathematical model for the pattern formation in desiccation cracking with emphasis on the emergence of the typical length scale with the typical geometry resulting from the hierarchical cell tessellation. The desiccation crack phenomenon is modeled as the coupling of desiccation, deformation, and fracture. This coupling model is numerically solved by weakly coupled analysis of the desiccation process and the deformation/fracture process. The basic features of the desiccation crack pattern and its formation process reproduced by the numerical analysis show reasonable agreement with experimental observations. This agreement implies that the proposed coupling model properly addresses the fundamental mechanism for the hierarchical pattern formation in desiccation cracking.

      PubDate: 2017-09-02T22:27:45Z
      DOI: 10.1016/j.physd.2017.08.002
       
  • Bifurcations of relative periodic orbits in NLS/GP with a triple-well
           potential
    • Authors: Roy H. Goodman
      Abstract: Publication date: Available online 5 August 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Roy H. Goodman
      The nonlinear Schrödinger/Gross–Pitaevskii (NLS/GP) equation is considered in the presence of three equally-spaced potentials. The problem is reduced to a finite-dimensional Hamiltonian system by a Galerkin truncation. Families of oscillatory orbits are sought in the neighborhoods of the system’s nine branches of standing wave solutions. Normal forms are computed in the neighborhood of these branches’ various Hamiltonian Hopf and saddle–node bifurcations, showing how the oscillatory orbits change as a parameter is increased. Numerical experiments show agreement between normal form theory and numerical solutions to the reduced system and NLS/GP near the Hamiltonian Hopf bifurcations and some subtle disagreements near the saddle–node bifurcations due to exponentially small terms in the asymptotics.

      PubDate: 2017-09-02T22:27:45Z
      DOI: 10.1016/j.physd.2017.07.007
       
  • A modified hybrid Van der Pol–Duffing–Rayleigh oscillator for
           modelling the lateral walking force on a rigid floor
    • Authors: Prakash Kumar; Anil Kumar; Silvano Erlicher
      Abstract: Publication date: Available online 4 August 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Prakash Kumar, Anil Kumar, Silvano Erlicher
      The paper proposes a single degree of freedom oscillator in order to accurately represent the lateral force acting on a rigid floor due to human walking. As a pedestrian produces itself the energy required to maintain its motion, it can be modelled as a self-sustained oscillator that is able to produce: (i) self-sustained motion; (ii) a lateral periodic force signal; and (iii) a stable limit cycle. The proposed oscillator is a modification of hybrid Van der Pol–Duffing–Rayleigh oscillator, by introducing an additional nonlinear hardening term. Stability analysis of the proposed oscillator has been performed by using the energy balance method and the Lindstedt–Poincare perturbation technique. Model parameters were identified from the experimental force signals of ten pedestrians using the least squares identification technique. The experimental and the model generated lateral forces show a good agreement.

      PubDate: 2017-09-02T22:27:45Z
      DOI: 10.1016/j.physd.2017.07.008
       
  • Nanopteron solutions of diatomic Fermi-Pasta–Ulam-Tsingou lattices
           with small mass-ratio
    • Authors: Aaron Hoffman; J. Douglas Wright
      Abstract: Publication date: Available online 31 July 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Aaron Hoffman, J. Douglas Wright
      Consider an infinite chain of masses, each connected to its nearest neighbors by a (nonlinear) spring. This is a Fermi-Pasta–Ulam-Tsingou lattice. We prove the existence of traveling waves in the setting where the masses alternate in size. In particular we address the limit where the mass ratio tends to zero. The problem is inherently singular and we find that the traveling waves are not true solitary waves but rather “nanopterons”, which is to say, waves which asymptotic at spatial infinity to very small amplitude periodic waves. Moreover, we can only find solutions when the mass ratio lies in a certain open set. The difficulties in the problem all revolve around understanding Jost solutions of a nonlocal Schrödinger operator in its semi-classical limit.

      PubDate: 2017-08-03T21:43:26Z
      DOI: 10.1016/j.physd.2017.07.004
       
  • Emergence of unstable modes for classical shock waves in isothermal ideal
           MHD
    • Authors: Heinrich Freistühler; Felix Kleber; Johannes Schropp
      Abstract: Publication date: Available online 31 July 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Heinrich Freistühler, Felix Kleber, Johannes Schropp
      This note studies classical magnetohydrodynamic shock waves in an inviscid fluidic plasma that is assumed to be a perfect conductor of heat as well as of electricity. For this mathematically prototypical material, it identifies, mainly numerically, two critical manifolds in parameter space, across which slow resp. fast MHD shock waves undergo emergence of a complex conjugate pair of unstable transverse modes. For slow shocks, this emergence occurs in a particularly interesting way already in the parallel case, in which it happens at the spectral value λ ˆ ≡ λ ∕ ω = 0 and the critical manifold possesses a simple explicit algebraic representation. Results of refined numerical treatment show that within the set of non-parallel slow shocks the unstable mode pair emerges from two generically different spectral values λ ˆ = ± i γ . For fast shocks, the critical manifold does not intersect the parallel regime and the emergence within the set of non-parallel fast shocks again starts from two generically different spectral values.

      PubDate: 2017-08-03T21:43:26Z
      DOI: 10.1016/j.physd.2017.07.005
       
  • Time-dependent spectral renormalization method
    • Authors: Justin T. Cole; Ziad H. Musslimani
      Abstract: Publication date: Available online 29 July 2017
      Source:Physica D: Nonlinear Phenomena
      Author(s): Justin T. Cole, Ziad H. Musslimani
      The spectral renormalization method was introduced by Ablowitz and Musslimani (2005), as an effective way to numerically compute (time-independent) bound states for certain nonlinear boundary value problems. In this paper, we extend those ideas to the time domain and introduce a time-dependent spectral renormalization method as a numerical means to simulate linear and nonlinear evolution equations. The essence of the method is to convert the underlying evolution equation from its partial or ordinary differential form (using Duhamel’s principle) into an integral equation. The solution sought is then viewed as a fixed point in both space and time. The resulting integral equation is then numerically solved using a simple renormalized fixed-point iteration method. Convergence is achieved by introducing a time-dependent renormalization factor which is numerically computed from the physical properties of the governing evolution equation. The proposed method has the ability to incorporate physics into the simulations in the form of conservation laws or dissipation rates. This novel scheme is implemented on benchmark evolution equations: the classical nonlinear Schrödinger (NLS), integrable P T symmetric nonlocal NLS and the viscous Burgers’ equations, each of which being a prototypical example of a conservative and dissipative dynamical system. Numerical implementation and algorithm performance are also discussed.

      PubDate: 2017-08-03T21:43:26Z
      DOI: 10.1016/j.physd.2017.07.006
       
 
 
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